Determining the position of an object with known distances to other objects having known positions is a conventional subject in the field of localization. In general, use is made of detection stations whose position is perfectly known, or “anchors”. In this case, the uncertainty in the result of the position lies mainly in the precision of the distance measurements, called “ranging” further below.
Another possible approach consists in determining the relative positioning of the nodes of a network of stations, on the sole basis of information of relative distances between all or some of these nodes. Precise knowledge of a limited amount of a priori position information regarding certain nodes then makes it possible to determine the absolute position of all of the nodes of the network. However, several obstacles have to be overcome, for example:                All of the distance measurements between the nodes are not necessarily accessible, due to the range of the ranging modules,        The distance measurements are necessarily affected by errors, making geometric resolution potentially overdetermined,        The information necessary for absolute positioning may be distributed over the coordinates of a plurality of nodes,        
Finally, the information necessary for absolute positioning may also be affected by errors, which are potentially larger than those associated with the ranging.
This last point is particularly important in practice, notably in collaborative navigation. Specifically, the absolute positioning information of certain nodes is often obtained by receivers of the satellite positioning system known under the abbreviation commercial GNSS (Global Navigation Satellite System), whose positioning error may, in difficult reception conditions, greatly exceed the inter-node distance measurement (ranging) errors. Ignoring this situation may then lead to positioning errors for free nodes, or even to the rejection of ranging measurements that are valid but incompatible with the distances calculated on the basis of the absolute positions, which are assumed to be error-free.
One of the technical problems posed is to find an effective solution for calculating the positions of nodes, so that a metric of the deviations between the ranging measurements and the inter-node distances of the solution is minimal, and while taking into account the uncertainty associated with the ranging errors and with the absolute positioning errors.
The prior art mainly describes two approaches to deal with this problem.
A first approach consists in reconstructing a relative geometry of the nodes of a network using ranging information and, possibly, inter-node angular information, for example by goniometry within a node. These methods are generally grouped together under the term “anchor-free localization”. In this approach, which uses only ranging measurements, all of the positions of the nodes are free, a priori, and the methods seek to adjust them such that the mean squared error between the calculated inter-node distances and the ranging measurements is minimal. A great number of algorithms have been proposed, from the simple mass-spring algorithm to more complex optimization techniques, for example, the least-squares solution by Newton-Raphson, particle filtering, etc. Most of the variants of this approach seek to avoid convergence to a local minimum, without however guaranteeing this.
A second approach aims to reconstruct the absolute position of the nodes of the network using ranging information through knowledge a priori of the position of some of said nodes, these being called “anchors”. This also involves minimizing the mean squared deviation between the distance between the positions of the nodes and the ranging measurements, certain positions being a priori fixed and known. The problem is then a problem of minimizing a function of the coordinates of free nodes, for example via a steepest descent algorithm. The article “Anchor-Based Three-Dimensional Localization Using Range Measurements”, Wang, Yue et al., illustrates this approach, which does not however guarantee that a global optimum is achieved. Some more theoretical works propose global optimization methods.
A third approach uses external sources or anchors having unknown positions. The problem is known under the term “self-calibration”, in which it is sought to jointly locate sources and the sensors that detect them. Time differences of arrival or TDOA are used, this giving less rich information than a direct ranging measurement, similar to a time of arrival or TOA, and not utilizing information a priori on the positions of the sources. The article “Calibration-Free TDOA Self-Localization, Wendeberg Johannes et al., in Journal of Location Based Services, May 2013”, illustrates one possible implementation of this approach through a plurality of conventional algorithms (mass-spring, gradient descent, Gauss Newton, etc.) or more specific algorithms (Cone alignment algorithm), these algorithms being known to those skilled in the art.
One of the drawbacks of the prior art is that it assumes:                Either that all of the positions of the sensors are free, delivering only a geometry of all of the nodes that is potentially precise, but relative coordinates of the nodes with respect to one another,        Or that the determination of the absolute position of the nodes is based on external knowledge of the position of some of said nodes whose accuracy is assumed to be perfect.        
A priori, the prior art does not deal with joint integration of the uncertainty as to the position of the anchors and the possible questioning of these positions.
FIG. 1 illustrates the problems that are not solved by the prior art. In this example, the coordinates of the anchors associated with the nodes M1 and M2 are assumed to be accessible ideally, that is to say without error, for example by virtue of topographical surveys, whereas the coordinates of the anchor M3 are in practice affected by an error d3, for example measured by a GNSS system, positioning the anchor M3 at the point M′3. Another assumption of this illustration is that the ranging precision is perfect, in practice much worse than d3. The ranging measurements are represented by arrows in FIG. 1. The prior art techniques incorrectly estimate the position of the free node at M′4, the actual position of the free node M4 by distributing the ranging errors associated with this node, which results in a significant residual mean squared error minimum. Furthermore, the position of the node M3 is not questioned, and remains erroneous.